Poincaré–Pontryagin–Melnikov Functions for a Type of Perturbed Degenerate Hamiltonian Equations

In this paper we consider polynomial perturbations of a family of polynomial Hamiltonian equations whose associated Hamiltonian is not transversal to infinity, and its complexification is not a Morse polynomial. We look for an algorithm to compute the first non-vanishing Poincaré–Pontryagin–Melnikov function of the displacement function associated with the perturbed equation. We show that the algorithm of the case when the Hamiltonian is transversal to infinity and its complexification is a Morse polynomial can be extended to our family of perturbed equations. We apply the result to study the maximum number of zeros of the first non-vanishing Poincaré–Pontryagin–Melnikov function associated with some perturbed Hamiltonian equations.


Introduction
Consider the perturbed Hamiltonian differential equation where F = F(x, y) is a real polynomial, and ω(ε) = A(x, y, ε)dx + B(x, y, ε)dy is a 1-form on the real plane R 2 such that A(x, y, ε) and B(x, y, ε) are real poly-B Salomón Rebollo-Perdomo srebollo@ubiobio.cl 1 Departamento de Matemática, Universidad del Bío-Bío, Avda. Collao 1202, Concepción, Chile nomials in x and y with coefficients that depend analytically on the real parameter ε ∈ (R, 0). Suppose that the foliation defined by the Hamiltonian differential equation (1 0 ) has at least a continuous family of cycles (periodic orbits) γ c ⊂ F −1 (c). As is widely known [19,26,27], for ε small enough and non-zero, (1 ε ) can have limit cycles bifurcating from the cycles γ c of (1 0 ). A common tool for studying these limit cycles is the displacement function associated with (1 ε ) and the family {γ c }: The coefficient L i (c) is the i-th order Poincaré-Pontryagin-Melnikov (PPM) function. The limit cycles of (1 ε ) that bifurcate from cycles of (1 0 ) are studied through the zeros of the first non-vanishing PPM function L k (c) with k ≥ 1. Indeed, the maximum number of isolated zeros, counting multiplicities, of L k (c) is an upper bound for the number of limit cycles of (1 ε ) that bifurcate from the cycles γ c of (1 0 ); in addition, the number of distinct zeros of L k (c) can provide a lower bound for the number of these limit cycles. See for instance Proposition 26.1 and Remark 26.2 in [22]. Hence, we need a mechanism for determining such a first non-vanishing function.
The classical Poincaré-Pontryagin formula says that the first order PPM function is given by an Abelian integral; however, if that function vanishes identically, then the PPM functions of order 2, 3, . . . have to be studied until either finding the first non-vanishing PPM function, or concluding that (1 ε ) is integrable.
It is known that for F which is transversal to infinity and whose complexification is a Morse polynomial, the first non-vanishing PPM function of the displacement function associated with (1 ε ) is an Abelian integral. See for instance Theorems 26.7 and 26.52 in [22]. The purpose of this paper is to show that this property remains true for a larger family of polynomials F and to apply the obtained result for studying the maximum number of zeros of the first non-vanishing PPM function of some equations (1 ε ).
In the present paper we will assume that the polynomial F defining (1 0 ) is of the form F = P(H ), where P = P(z) is a univariate polynomial of degree greater than one, and H = H (x, y) is a bivariate polynomial. Under these assumptions F is not transversal to infinity, and its complexification, by considering x and y as complex variables, is not a Morse polynomial because F has degenerate singularities at H −1 (Z (P )), where P = P (z) denotes the derivative of P(z) with respect to z, and Z (P ) denotes the finite set of (complex) solutions of P (z) = 0.
Since the coefficients of A(x, y, ε) and B(x, y, ε) depend analytically on ε and d F = P (H )d H, we can rewrite (1 ε ) as where ω i = A i (x, y)dx + B i (x, y)dy, for i = 1, 2, . . ., is a polynomial 1-form.
We are assuming that deg(P) ≥ 2, then P (H ) is not a constant polynomial; this implies that the complexification of the unperturbed system (3 0 ), i.e., P (H )d H = 0 has curves of singularities given by H −1 (z), with z such that P (z) = 0. Thus, we say that (3 0 ) is a degenerate Hamiltonian differential equation and (3 ε ) is a perturbed degenerate Hamiltonian differential equation. Moreover, when P (z) has real solutions, then in R 2 \H −1 (Z (P )) the perturbed equation (3 ε In addition, if {γ c } is a family of cycles of (3 0 ), then it does not intersect H −1 (Z (P )). Hence, as in the complement of Z (P ) the polynomial P is a local invertible map, we can reparametrize {γ c } by using z = H | {γ c } . We then have a family {γ z }; furthermore, γ c and γ z are the same cycle if c = P(z). Therefore, we can write the displacement function associated with (3 ε ) in terms of ε and z instead of ε and c.
The main motivation for studying (3 ε ) is that there is a greater variety of bifurcation phenomena of limit cycles in perturbation of systems with curves of singularities than when the unperturbed system has only isolated singularities. For example more limit cycles can bifurcate in the former case than in the latter one. This property has been shown in many papers by studying systems of the form Note that in such a case d H = 0 is the linear center, the simplest planar Hamiltonian equation having a family of cycles. The following cases for the set of singularities {G = 0} have been considered: one straight line in [24]; two parallel straight lines in [30]; two orthogonal lines in [4]; three lines, two of them parallel and one perpendicular, in [6]; four lines with a special configuration in [1]; any quadratic algebraic curve, with deg ω 1 ≤ 3, in [17] and some quadratic algebraic curves, with deg ω 1 ≤ n, in [23]; one multiple straight line of singularities in [11,31]; K isolated singular points in [10]; K straight lines that are parallel to the coordinate axes in [12].
In light of these recent developments it is interesting to study the case when {G = 0} is formed by one or several level curves of H , that is, G(x, y) = P (H ) for a univariate polynomial P (z). For instance, system (3 ε ) with P (z) = z p and with H = (x 2 + y 2 )/2 was considered in [3], where the authors give upper bounds for the maximum number of zeros, counting multiplicities, of the first non-vanishing PPM function and show that if p ≥ 1 then L k (z) with 2 ≤ k ≤ 3 has more zeros than for the case p = 0 which was studied by Iliev in [18].
In the present paper we generalize the results of [3] by considering P(z) as an arbitrary polynomial of degree p + 1. Our result is the following.
(a) For each k ∈ {1, 2}, each p ≥ 1, and each n ≥ 2 there exists a polynomial P(z) of degree p + 1 and a polynomial perturbation of degree n of (3 0 ) such that L k (z) has exactly Z k (n, p) simple zeros in (0, ∞)\Z (P ). (b) For k = 3, each p ∈ {1, 2}, and each n ≥ 2, there exists a polynomial P(z) of degree p + 1 and a polynomial perturbation of degree n of (3 0 ) such that L 3 (z) has exactly Z 3 (n, p) simple zeros in (0, ∞)\Z (P ).
Moreover, Theorem 1 also improves some results of [3] because we prove that the upper bounds for L 3 (z) are reached in cases p = 1 and p = 2. Tables 1 and 2 give the numerical values of Z k (n, p) for k = 1, 2, 3 with p = 1 and p = 2, respectively.
The proof of Theorem 1 will be obtained as a consequence of our study of (3 ε ) in a more general setting. More precisely, we will prove that under generic conditions on H we can provide an algorithm to compute the first non-vanishing PPM function L k (z), with k ≥ 2, of the displacement function of (3 ε ); in particular we obtain that L k (z) is an Abelian integral. Of course, we know that L 1 (z) is an Abelian integral: the integral of ω 1 along the cycles γ z . However, when L 1 (z) ≡ 0 for (3 ε ) the Francoise's algorithm [8,18] can not be applied to calculate L 2 (z) because we would need that ω 1 = d Q + qd F. This condition can not be ensured by the Ilyashenko-Gavrilov theorem [13,19] (see Theorem 3 in Section 2) since F does not have isolated singular points. Hence, a new formula for L k (z) with k ≥ 2 must be obtained for the perturbed degenerate Hamiltonian differential equation (3 ε ). Our main result in this sense is the next one.

Theorem 2 Assume that H is transversal to infinity and its complexification is a
Morse polynomial. If L k (z) with k ≥ 2 is the first non-vanishing PPM function of the displacement function associated with (3 ε ), then there are polynomials q 1 , . . . , q k−1 , Q 1 , . . . , Q k−1 , q, and Q, with q ≡ Q ≡ 0 if k = 2, such that and We observe that if the degree of P is one, then we can assume without loss of generality that P = 1, which implies that R i = 0 for i ≥ 2. In such a case Theorem 2 reduces to the formula given by Iliev [18,Lemma 2] for F = (x 2 + y 2 )/2. Hence, Iliev's formula and Theorem 2 generalize the results of Françoise [8] and Yakovenko [33].
We point out that following mainly the ideas of Ilyashenko [19], Gavrilov [13], Iliev [18], and Françoise [8] it is possible to provide a more compact formula for L k (c) (see Proposition 1 and (9) in Section 3) than the one given in Theorem 2. However, the upper bounds for the number of zeros of L k (z) obtained in terms of the degree of the 1-form defining the compact formula may not always be attained, and in this sense such a compact formula is not optimal. After a detailed analysis we can transform the compact formula to the formula of Theorem 2. This new formula, although more tedious, is very useful because it provides tighter upper bounds for the number of zeros of L k (z). For example, in the case H = (x 2 + y 2 )/2 the exact upper bound for k = 2, 3 is obtained.
We know [22,Theorem 26.3] that the first order PPM function in (2) is given by the Abelian integral L 1 (c) = γ c ω 1 , which can be treated and understood in a better way if we consider it on the complex plane C 2 since its analytic continuation depends on the singular values and on the monodromy of the complexification of F. Moreover, although the study of limit cycles of planar differential equations was originally stated on R 2 , one can consider it on C 2 [13][14][15][16]19,21,25,32] because a real (limit) cycle of (1 ε ) is a complex (limit) cycle of its complexification [20, p. 340]. Hence we can investigate (1 ε ) on C 2 . This paper is organized as follows. The background and definitions are delivered in Sect. 2. In Sect. 3 we will study (3 ε ) in the complex setting and we will derive the relationship between the PPM functions of the displacement functions associated with (3 ε ) and (4 ε ), respectively. The proof of Theorem 2 will be given in Sect. 4. In Sect. 5 we will study (3 ε ) with H = (x 2 + y 2 )/2 on the complex plane, and finally in Sect. 6 we will give the proof of Theorem 1.

Background and Definitions
From now on we will consider F as a polynomial in the ring C[x, y] of complex polynomials in two complex variables with coefficients in C, ε as a small complex parameter, and ω as a complex 1-form, i.e. we think A(x, y, ε) and B(x, y, ε) as complex polynomials in x and y with coefficients that depend analytically on ε ∈ (C, 0).
For studying the (complex) limit cycles of (1 ε ) on the complex plane that bifurcate from (1 0 ) we have to recall some definitions and some properties of the elements that define the Eq. (1 ε ). For that, we will divide this section into three parts. In the first one, we will provide the notion of cycle and limit cycle in the complex setting, as well as the definition of the holonomy map, which is the complexification of the well-known first return map. The second part is devoted to recalling some properties of complex polynomials in order to know the structure of the foliation defined by the Hamiltonian equation (1 0 ), and in the third one we will give some definitions related to complex polynomial 1-forms, which link their analytical or algebraic properties with respect to the Hamiltonian equation (1 0 ).

Complex Cycle, Complex Limit Cycle, and the Holonomy Map
We know that for each ε ∈ (C, 0) the differential equation (1 ε ) defines a 1-dimensional complex foliation with singularities F ε on C 2 : F ε is a foliation by Riemann surfaces with singularities on C 2 . Definition 1 Consider a leaf of F ε . Let γ ⊂ be a real curve homeomorphic to the unit circle S 1 , and let [γ ] be its free homotopy class on . If γ is not homotopic to a point on , then [γ ] is called a complex cycle of (1 ε ).
To give the definition of complex limit cycle, we need to recall the construction of the holonomy map, which is as follows. We consider a complex cycle [γ ] of (1 ε 0 ). Let U be an annular neighborhood of γ in , and let V be a tubular neighborhood of U in Let p 0 be a point of γ and consider a parametrization γ : [0, 1] → of γ such that p 0 = γ (0) = γ (1) (we are identifying γ (t) with its image γ ). Because of the commutativity of the previous diagram, the set L : , the curve γ may be lifted to a unique curve γ (ε, p) : [0, 1] → C 2 that lies on the leaf (ε, p) passing through (ε, p) and that covers γ under the retraction π . In other words, and it is analytic because of the analytic dependence of the leaves of {F ε } on initial conditions.
We can identify the transversal L with (C, ε 0 ) × D by using the parametrization The holonomy map associated to γ is then defined by the analytic map

Remark 2
The map f γ does not depend, up to analytic conjugation, on the representative of [γ ], the parametrization of the representative, the point p 0 , or the transversal section L (i.e. the retraction π and the biholomorphism ψ). This property implies that if γ is another representative of [γ ] and γ (δ, c) is the corresponding holonomy map, then γ (δ, c) and γ (δ, c) are analytically conjugated.
Remark 3 Any real (limit) cycle γ of the real differential equation (1 ε 0 ) defines a complex (limit) cycle [γ ] of the complexification of (1 ε 0 ), and the holonomy map γ (ε, c) is the complexification of the first return map associated with γ .

Properties of Complex Polynomials
It is well-known that for each F ∈ C[x, y] there is a finite set F ⊂ C such that is a locally trivial smooth fibration. See for instance [2]. This finite set F is the set of singular values of F, which is composed of the values in C coming from the finite singular points of F and of the "singular points at infinity" of F [7]. Every value c ∈ C\ F is a generic value of F and the corresponding fiber which is an affine nonsingular algebraic curve, is a generic fiber of F. A complex bivariate polynomial is primitive if its generic fiber is irreducible; in such a case, the generic fiber is diffeomorphic to a compact Riemann surface of genus g ≥ 0 punctured at h ≥ 1 different points: the points at infinity of the fiber, and the polynomial is called type (g, h) [29].
We know [9] that for each F ∈ C[x, y] there are a primitive polynomial H ∈ C[x, y] and a univariate polynomial P ∈ C[z], such that F = P(H ). Note that if F = P(H ) and P is of degree one, then F is primitive. Thus, the Hamiltonian foliation F 0 = {d F = 0} defined by (1 0 ), which is given by the fibers of F, is a foliation by punctured compact Riemann surfaces of finite genus with singularities on C 2 .
If F ∈ C[x, y] has degree n + 1 ≥ 2, then it is transversal to infinity, if its homogeneous part of degree n + 1 factors out as the product of n + 1 pairwise different linear forms; furthermore, F is a Morse polynomial, if it has n 2 singular points and n 2 singular values.

Properties of Complex Polynomial 1-Foms
Here we will recall two properties that a complex polynomial 1-form ω can have with respect to the foliation F 0 . Such properties will be useful in the study of the higher order PPM functions of the displacement function associated with (1 ε ).
Definition 3 A 1-form ω is analytically relatively exact with respect to F 0 if for each value c ∈ C and each homological cycle δ ∈ H 1 (F −1 (c), Z) the integral of ω along δ is zero: Clearly each polynomial 1-form that is algebraically relatively exact is also analytically relatively exact with respect to F 0 . The inverse connection between these two properties is given by the following result.
Theorem 3 (Ilyashenko [19], Gavrilov [13]) Suppose that F ∈ C[x, y] has isolated singular points in C 2 and that F −1 (c) is connected for each c ∈ C. A complex polynomial 1-form ω is then algebraically relatively exact with respect to F 0 if and only if it is analytically relatively exact with respect to it.

Perturbed Degenerate Hamiltonian Equations
We suppose that the polynomial F(x, y) defining the Hamiltonian equation (1 0 ) is non-primitive; hence F = P(H ) for a univariate polynomial P(z) of degree r ≥ 2 and a bivariate primitive polynomial H (x, y). Thus, we have the perturbed degenerate Hamiltonian differential equation (3 ε ).

The Complex Displacement Function
Let c 0 be a generic value of F, and consider a fixed complex cycle [γ c 0 ] of (1 0 ). We take γ := γ c 0 as a representative loop of [γ c 0 ]. We can transport this loop continuously into the neighboring fibers according to the fibration (5) The complex displacement function associated with (1 ε ) and γ is defined as Since F = P(H ), there are r ≥ 2 different generic values z 1 , . . . , z r of H such that the generic fiber F −1 (c 0 ) of F is the disjoint union of the r generic fibers L z 1 , . . . , L z r of H . L z i , for each i = 1, . . . , r , is diffeomorphic to a finite punctured compact Riemann surface of finite genus. Hence, the loop γ defining the fixed cycle [γ c 0 ] is contained in one of the fibers L z 1 , . . . , L z r of H . Without loss of generality we can suppose that γ ⊂ L z 1 ; thus, every transported loop γ c is contained in a generic fiber L z of H , where z varies in the neighborhood D(z 1 ) := P −1 (D(c 0 )) of z 1 which is contained in the set of generic values of H . We can then reparametrize the family {γ c } by using z, as a result we have the family {γ z } with z ∈ D(z 1 ). Thus, analogous to the previous construction, the complex displacement function associated with (4 ε ) and γ is defined as

Poincaré-Pontryagin-Melnikov Functions
By construction, the displacement function L F,γ (ε, c) is analytic, and it admits a power series In addition, since we have the parametrization P : D(z 1 ) → D(c 0 ), we can write L F,γ (ε, c) in terms of (ε, z); thus we have Analogously, Of course, if k ≥ 1 and L k (z) vanishes identically on D(z 1 ), then L k (c) vanishes identically on D(c 0 ). In addition, if L k (z) is the first non-vanishing coefficient in (7), then its maximum number of zeros in D(z 1 ) coincides with the maximum number of zeros in D(c 0 ) of the first non-vanishing coefficient L k (c) in (6). Furthermore, we know that the first non-vanishing PPM function of L F,γ (ε, z) depends only on the free homotopy class [γ z ] of γ z . See [16].
Hence P (ζ (ε, z)) can be written as P (ζ (ε, z)) = P (z) + ε ζ 1 (z) + ε 2 ζ 2 (z) + · · · , whereby we obtain Thus, if L k (z) is the first non-vanishing PPM function of L F,γ (ε, z) then with l k (z) being the first PPM function of L H,γ (ε, z) that does not vanish identically. Hence, for determining the form of L k (z) in terms of the perturbation of (3 0 ) we will obtain a recursive formula for l k (z). The Poincaré-Pontryagin formula says that the first coefficient in (8) is given by the integral of ω 1 /P (H ) along γ z , and as P (H ) is constant on γ z it follows that To compute the first coefficient l k (z) in (8) that does not vanish identically with k ≥ 2, we construct inductively a sequence of polynomial 1-forms by assuming that the polynomial H has only isolated singular points in C 2 and that the fiber L z of H is connected for each z ∈ C.
The construction of the sequence is as follows.
The following two propositions will allow us to obtain the formula for l k (z), and as a result the expression of L k (z).

Proposition 2 Suppose k ≥ 2.
If ω k is the first non analytically relatively exact 1form in the sequence ω 1 , . . . , ω k−1 , ω k constructed inductively by (11), then there are polynomials q and Q, with q = Q ≡ 0 if k = 2, such that and with R 1 := 1/P and R i := R 1 R i−1 for i ≥ 2.
The proof of this proposition will be given at the end of next section.

Lemma 1
If Proof The result follows because a simple computation shows that where

Proof of Theorem 2
We will prove Theorem 2 by assuming Proposition 2.

Proof of Theorem 2
By assumption H is a real polynomial transversal to infinity and its complexification is a Morse polynomial, then from [22,Theorem 26.52] it follows that we can consider the sequence ω 1 , . . . , ω k according to (11). Moreover, the polynomials q 1 , . . . , q k−1 and Q 1 , . . . , Q k−1 involved in the construction of the sequence are real polynomials because H and ω i with i ≥ 1 are real objects. Analogously, the polynomials q and Q in the definition of ω k3 (as in (13)) are real. From (9) we know that L k (z) = P (z)l k (z), and from Proposition 1 we have Moreover, by assuming Proposition 2 it follows that Therefore, by applying Lemma 1 we obatain where ω k1 , ω k2 , and ω k3 are as in (12) and (13), which are precisely the expressions given in the theorem.

Proof of Proposition 2
We proceed by induction on k. The base case is k = 2; in such a case, from (12) and (13) it follows that ω 21 = P ω 2 , ω 22 = 0, and ω 23 = R 1 P q 1 ω 1 = q 1 ω 1 because R 1 P = (1/P )P = 1. Moreover, ω 1 = d Q 1 + q 1 d H for some polynomials Q 1 and q 1 . Then Thus, as R 2 = R 1 R 1 = (1/P )(1/P ) = −P /(P ) 3 we obtain Hence, the proposition is true for k = 2. Now, we assume that the result is true for k − 1, and we will prove it for k. By using (12) it is easy to see that ω k , given in (11), can rewritten as If we prove that then We then obtain which proves the result for k. Therefore, to complete the proof remains to prove that (16) holds. By using Ilyashenko-Gavrilov theorem and straightforward computations it is possible to prove the assertion.
The following lemma will be useful for obtaining upper bounds for the number of zeros of the Poincaré-Pontryagin-Melnikov functions.
Proof Let P = P (z) and R i = R i (z). Since P R 1 = P (1/P ) = 1, the result holds for k = 1. Now, we assume that the result is true for k − 1, and we will prove it for k. By the induction hypothesis, (P ) 2 As

PPM Functions of (3 ε ) with H = 1 2 (x 2 + y 2 )
Along this section we will consider (3 ε ) in the complex plane and H will denote the polynomial H (x, y) = (x 2 + y 2 )/2 : C 2 → C. This polynomial has the following properties: the origin of C 2 is the unique singular point of H , the fiber L 0 = H −1 (0) is the connected union of the two complex lines {x − √ −1y = 0} and {x + √ −1y = 0}, and for z = 0 the map is a parametrization of L z . Thus, H satisfies the hypothesis of the Ilyashenko-Gavrilov theorem. Hence, we can consider the sequence of polynomial 1-forms ω 1 , ω 2 , . . . , constructed inductively in (11) with H = (x 2 + y 2 )/2. Moreover, if we assume that ω k with k ≥ 2 is the first non analytically relatively exact 1-form in the sequence ω 1 , ω 2 , . . . , ω k , then from Proposition 2 it follows that and Moreover, the polynomials q 1 , . . . , q k−1 and Q 1 , . . . , Q k−1 satisfy Next, we will state a result about the degree of ω i for i = 1, 2, . . . , k and another about the degree of ω k1 , ω k2 , and ω k3 , which will be useful in the proof of Theorem 1. We will prove them later on.

Proposition 4
If deg ω i ≤ n for i ≥ 1, deg P = p + 1, and ω k with k ≥ 2 is the first non analytically relatively exact 1-form in the sequence ω 1 , ω 2 , . . . , ω k , then In the study of PPM functions related to (3 ε ), as well as in the proof of the previous two propositions, is essential the following result, which give us the structure of a polynomial 1-form with respect to the Hamiltonian H .
Since H is transversal to infinity and it is a Morse polynomial, the first non-vanishing PPM function L k (z) of the displacement function associated with (3 ε ) is an Abelian integral because of Theorem 2. This property implies that L k (z) depends only on the homological cycles of L z . By abusing of the notation we will denote by [γ z ] the homological class of a loop γ z in L z .
As L z is homeomorphic to C\{0}, then it has only one nontrivial homological cycle. The homological cycle [γ z ] with γ z = ϕ z (α), where α is the unit circle in C\{0} is the generator of H 1 (L z , Z).
By assuming Propositions 3, 4, and 5 we will now prove the next theorem, which gives the maximum number of isolated zeros of the first non-vanishing PPM function associated with (3 ε ).

Theorem 4 Let n = sup{deg(ω i ) | i ≥ 1} and p + 1 = deg P(z). If L k (c) is the first non-vanishing PPM function of the displacement function associated with (3 ε ), then an upper bound for the maximum number of isolated zeros in (C\{0})\Z (P ), counting multiplicities, of L k (c) is
Proof Firstly we assume that k = 1. From (9) and (10) we then obtain the Poincaré-Pontryagin formula: From Proposition 5 it follows that ω 1 = d Q 1 +q 1 d H + q 1 (H )ydx with q 1 a univariate polynomial of degree at most [(deg ω 1 −1)/2], where deg ω 1 is the degree of ω 1 . Thus, By using the parametrization ϕ z (t) given by (17) we have This implies that the degree of q 1 is an upper bound for the maximum number of isolated zeros, counting multiplicities, of L 1 (z) in C\{0}.
We now assume k ≥ 2. From Theorem 2 we know that where ω k1 , ω k2 , and ω k3 are as in (12) and (13). Furthermore, as because of Proposition 5, we then have Proposition 5 says that q k is of degree at most [(deg ω k − 1)/2]. Therefore, to complete the proof we must prove that Next we will prove this assertion.
The proof of the Propositions 3 and 4 follows by induction on k. We will give the proof of the former, the proof of the latter is analogous.
We now assume that the result is true for i ≤ k − 1, and we will prove it for k. We will split the proof in two parts: n ≤ 2 p + 1 and n > 2 p + 1.

Proof of Theorem 1
In this section we will consider (3 ε ) with H (x, y) = (x 2 + y 2 )/2 : R 2 → R. The results of previous section can then be restricted to this real case. Moreover, now γ z = H −1 (z) for z ∈ (0, ∞) is a circle, centering at the origin of R 2 of radius √ 2z.
Proof of Theorem 1 The first part of the theorem is a corollary of Theorem 4. Hence, proving the second part remains: statements (a) and (b). Proof of (a). The assertion follows easily for k = 1. Indeed, it is sufficient to consider (3 ε ) with P (z) = z p and ω 1 For k = 2 we will split the proof in two cases: n ≤ 2 p + 1 and n > 2 p + 1. Case 1. Assume n ≤ 2 p + 1. In (3 ε ) we take Thus, L 1 (z) ≡ 0, and from Theorem 2 we have By using the expressions of P , ω 1 , and ω 2 we obtain Hence, for a small enough, L 2 (z) has p + n−1 2 simple zeros in (0, ∞)\Z (P ). Case 2. Assume n > 2 p + 1. We consider (3 ε and i=0 d i H i . Thus, L 1 (z) ≡ 0 and straightforward computations show that in both cases n = 2m and n = 2m + 1 the second function L 2 (z) takes the form where G n−1 is a polynomial of degree Z 2 (n, p) = n − 1 with n independent coefficients, which implies that L 2 (z) can have Z 2 (n, p) simple zeros in (0, ∞)\Z (P ). Proof of (b). Again, we will consider two cases: n ≤ 2 p + 1 and n > 2 p + 1.